A third way to think is to identify the lambda formula not with classes of alphabetic sequences, but rather with abstract structures that we might draw like this:
<pre><code>
- λ ... ___ ...
- ^ |
- |______|
+ (λ. λ. _ _) y
+ ^ ^ | |
+ | |__| |
+ |_______|
</code></pre>
Here there are no bound variables, but there are *bound positions*. We can regard formula like (a) and (b) as just helpfully readable ways to designate these abstract structures.
Syntactic equality, reduction, convertibility
=============================================
-Define T to be `(\x. x y) z`. Then T and `(\x. x y) z` are syntactically equal, and we're counting them as syntactically equal to `(\z. z y) z` as well, which we will write as:
+Define N to be `(\x. x y) z`. Then N and `(\x. x y) z` are syntactically equal, and we're counting them as syntactically equal to `(\z. z y) z` as well, which we will write as:
-<pre><code>T ≡ (\x. x y) z ≡ (\z. z y) z
+<pre><code>N ≡ (\x. x y) z ≡ (\z. z y) z
</code></pre>
This:
- T ~~> z y
+ N ~~> z y
-means that T beta-reduces to `z y`. This:
+means that N beta-reduces to `z y`. This:
- M <~~> T
+ M <~~> N
-means that M and T are beta-convertible, that is, that there's something they both reduce to in zero or more steps.
+means that M and N are beta-convertible, that is, that there's something they both reduce to in zero or more steps.
Combinators and Combinatorial Logic
===================================
There are some well-known linguistic applications of Combinatory
Logic, due to Anna Szabolcsi, Mark Steedman, and Pauline Jacobson.
-Szabolcsi supposed that the meanings of certain expressions could be
-insightfully expressed in the form of combinators.
-
+They claim that natural language semantics is a combinatory system: that every
+natural language denotation is a combinator.
For instance, Szabolcsi argues that reflexive pronouns are argument
duplicators.
Notice that the semantic value of *himself* is exactly `W`.
-The reflexive pronoun in direct object position combines first with the transitive verb (through compositional magic we won't go into here). The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.
+The reflexive pronoun in direct object position combines with the transitive verb. The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.
Note that `W <~~> S(CI)`:
That is, K throws away its second argument. The reduction rule for S can be constructed by examining
the defining lambda term:
- S = \fgx.fx(gx)
+<pre><code>S ≡ \fgx.fx(gx)</code></pre>
S takes three arguments, duplicates the third argument, and feeds one copy to the first argument and the second copy to the second argument. So:
So the combinator `SKK` is equivalent to the combinator I.
-Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. The most common system uses S, K, and I as defined here.
+Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. As we said, the most common system uses S, K, and I as defined here.
###The equivalence of the untyped lambda calculus and combinatory logic###
[Fussy notes: if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of S, K, and I (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of our boolean false `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In the intermediate stage, we have `\x.I`, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.]
+[Various, slightly differing translation schemes from combinatorial logic to the lambda calculus are also possible. These generate different metatheoretical correspondences between the two calculii. Consult Hindley and Seldin for details. Also, note that the combinatorial proof theory needs to be strengthened with axioms beyond anything we've here described in order to make [M] convertible with [N] whenever the original lambda-terms M and N are convertible.]
+
+
Let's check that the translation of the false boolean behaves as expected by feeding it two arbitrary arguments:
KIXY ~~> IY ~~> Y
Throws away the first argument, returns the second argument---yep, it works.
-Here's a more elaborate example of the translation. The goal is to establish that combinators can reverse order, so we use the T combinator, where `T = \x\y.yx`:
+Here's a more elaborate example of the translation. The goal is to establish that combinators can reverse order, so we use the **T** combinator, where <code>T ≡ \x\y.yx</code>:
[\x\y.yx] = [\x[\y.yx]] = [\x.S[\y.y][\y.x]] = [\x.(SI)(Kx)] = S[\x.SI][\x.Kx] = S(K(SI))(S[\x.K][\x.x]) = S(K(SI))(S(KK)I)
We can test this translation by seeing if it behaves like the original lambda term does.
The orginal lambda term lifts its first argument (think of it as reversing the order of its two arguments):
- S(K(SI))(S(KK)I) X Y =
- (K(SI))X ((S(KK)I) X) Y =
- SI ((KK)X (IX)) Y =
- SI (KX) Y =
- IY (KX)Y =
- Y X
+ S(K(SI))(S(KK)I) X Y ~~>
+ (K(SI))X ((S(KK)I) X) Y ~~>
+ SI ((KK)X (IX)) Y ~~>
+ SI (KX) Y ~~>
+ IY (KXY) ~~>
+ Y X
-Viola: the combinator takes any X and Y as arguments, and returns Y applied to X.
+Voilà: the combinator takes any X and Y as arguments, and returns Y applied to X.
One very nice property of combinatory logic is that there is no need to worry about alphabetic variance, or
variable collision---since there are no (bound) variables, there is no possibility of accidental variable capture,
logic is that anything that can be done by binding variables can just as well be done with combinators.
This has given rise to a style of semantic analysis called Variable Free Semantics (in addition to
Szabolcsi's papers, see, for instance,
-Pauline Jacobson's 1999 *Linguistics and Philosophy* paper, `Towards a variable-free Semantics').
+Pauline Jacobson's 1999 *Linguistics and Philosophy* paper, "Towards a variable-free Semantics").
Somewhat ironically, reading strings of combinators is so difficult that most practitioners of variable-free semantics
-express there meanings using the lambda-calculus rather than combinatory logic; perhaps they should call their
+express their meanings using the lambda-calculus rather than combinatory logic; perhaps they should call their
enterprise Free Variable Free Semantics.
-A philosophical application: Quine went through a phase in which he developed a variable free logic.
+A philosophical connection: Quine went through a phase in which he developed a variable free logic.
- Quine, Willard. 1960. Variables explained away. {\it Proceedings of
- the American Philosophical Society}. Volume 104: 343--347. Also in
- W.~V.~Quine. 1960. {\it Selected Logical Papers}. Random House: New
+ Quine, Willard. 1960. "Variables explained away" <cite>Proceedings of the American Philosophical Society</cite>. Volume 104: 343--347. Also in W. V. Quine. 1960. <cite>Selected Logical Papers</cite>. Random House: New
York. 227--235.
The reason this was important to Quine is similar to the worries that Jim was talking about
-in the first class in which using non-referring expressions such as Santa Clause might commit
-one to believing in non-existant things. Quine's slogan was that `to be is to be the value of a variable'.
+in the first class in which using non-referring expressions such as Santa Claus might commit
+one to believing in non-existant things. Quine's slogan was that "to be is to be the value of a variable."
What this was supposed to mean is that if and only if an object could serve as the value of some variable, we
are committed to recognizing the existence of that object in our ontology.
Obviously, if there ARE no variables, this slogan has to be rethought.
Quine did not appear to appreciate that Shoenfinkel had already invented combinatory logic, though
he later wrote an introduction to Shoenfinkel's key paper reprinted in Jean
-van Heijenoort (ed) 1967 *From Frege to Goedel,
- a source book in mathematical logic, 1879--1931*.
+van Heijenoort (ed) 1967 <cite>From Frege to Goedel, a source book in mathematical logic, 1879--1931</cite>.
+
Cresswell has also developed a variable-free approach of some philosophical and linguistic interest
in two books in the 1990's.
A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is
-from combinatory logic (see especially his 2000 book, *The Syntactic Process*). Steedman attempts to build
-a syntax/semantics interface using a small number of combinators, including T = \xy.yx, B = \fxy.f(xy),
+from combinatory logic (see especially his 2000 book, <cite>The Syntactic Processs</cite>). Steedman attempts to build
+a syntax/semantics interface using a small number of combinators, including T ≡ `\xy.yx`, B ≡ `\fxy.f(xy)`,
and our friend S. Steedman used Smullyan's fanciful bird
names for the combinators, Thrush, Bluebird, and Starling.
\x. M
+When we add eta-reduction to our proof system, we end up reconstruing the meaning of `~~>` and `<~~>` and "normal form", all in terms that permit eta-reduction as well. Sometimes these expressions will be annotated to indicate whether only beta-reduction is allowed (<code>~~><sub>β</sub></code>) or whether both beta- and eta-reduction is allowed (<code>~~><sub>βη</sub></code>).
+
The logical system you get when eta-reduction is added to the proof system has the following property:
> if `M`, `N` are normal forms with no free variables, then <code>M ≡ N</code> iff `M` and `N` behave the same with respect to every possible sequence of arguments.
-That is, when `M` and `N` are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments <code>L<sub>1</sub>, ..., L<sub>n</sub></code> such that:
+This implies that, when `M` and `N` are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments <code>L<sub>1</sub>, ..., L<sub>n</sub></code> such that:
<pre><code>M L<sub>1</sub> ... L<sub>n</sub> x y ~~> x
N L<sub>1</sub> ... L<sub>n</sub> x y ~~> y
</code></pre>
-That is, closed normal forms that are not just beta-reduced but also fully eta-reduced, will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
+So closed beta-plus-eta-normal forms will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
So the proof theory with eta-reduction added is called "extensional," because its notion of normal form makes syntactic identity of closed normal forms coincide with extensional equivalence.
+See Hindley and Seldin, Chapters 7-8 and 14, for discussion of what should count as capturing the "extensionality" of these systems, and some outstanding issues.
+
The evaluation strategy which answers Q1 by saying "reduce arguments first" is known as **call-by-value**. The evaluation strategy which answers Q1 by saying "substitute arguments in unreduced" is known as **call-by-name** or **call-by-need** (the difference between these has to do with efficiency, not semantics).
* [Scooping the Loop Snooper](http://www.cl.cam.ac.uk/teaching/0910/CompTheory/scooping.pdf), a proof of the undecidability of the halting problem in the style of Dr Seuss by Geoffrey K. Pullum
+Interestingly, Church also set up an association between the lambda calculus and first-order predicate logic, such that, for arbitrary lambda formulas `M` and `N`, some formula would be provable in predicate logic iff `M` and `N` were convertible. So since the right-hand side is not decidable, questions of provability in first-order predicate logic must not be decidable either. This was the first proof of the undecidability of first-order predicate logic.
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